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Introduction

Assignment: Experiments, Events, and Outcomes

When a probability experiment is performed, there are many ways the outcomes can be divided into events. How these events are defined often determines how the probabilities are distributed and calculated. Students are not ready to know all the details this early in the course, but they should learn to be flexible and creative in defining events.

Definitions
A simple event is one that contains only one outcome, or in other words, can happen in only one way.
A compound event contains more than one outcome.

Example:

  1. For the experiment of rolling a six-sided die, there are six possible outcomes. The sample space, the set of all possible outcomes, can be represented as S = \left \{1, 2, 3, 4, 5, 6 \right \}. List two simple and two compound events for this experiment.

Answer: Simple Events: A = \left \{1 \right \} or \ B = \left \{5 \right \}

Compound \ Events:

C = {rolling an even number} = \left \{2, 4, 6 \right \} or

D = {rolling an number less than three} = \left \{1, 2 \right \}

Exercises:

  1. The experiment of drawing one card from a standard deck has 52 outcomes. Define two simple and two compound events.
  2. Buying a box of cereal is a probability experiment where every different type of cereal is a possible outcome. Define two simple and two complex events.
  3. Describe a probability experiment and the possible outcomes. Define two simple and two complex events.

Answer will vary.

Compound Events

Assignment: More Practice with Unions and Intersections

Students can always use more practice finding the intersection and union of sets. A good guideline for students to keep in mind is that the intersection of two events is more restrictive and results in a set that is smaller than, or in some cases equal in size to the original sets, and that the union of two events is more inclusive and results in a set that is larger than, or in some cases equal in size to the original sets.

Exercises:

1. Experiment: drawing one card from a standard deck of cards

Events: A = \left \{ \mathrm{a \ red \ suite}\right \}, \ B = \left \{ \mathrm{a \ face \ card}\right \} = \left \{\mathrm{Jack, \ Queen, \ King} \right \}

List the outcomes in A \cup B, and in A \cap B. If each card in the deck is equally likely to be chosen what is the probability of each compound event?

2. Experiment: Asking participants the question, “What is your favorite day of the week?”

Events: A = \left \{ \mathrm{their \ response \ is \ a \ weekday}\right \} = \left \{\mathrm{M, T, W, H , F} \right \},

B = \left \{\mathrm{their \ response \ is \ a \ weekend}\right \} = \left \{\mathrm{S, U} \right \}

Find A \cup B and A \cap B.

3. Describe a probability experiment and events such that the intersection of the events is the empty set and the union of the events is the entire sample space.

Answers:

1. A \cap B  = \left \{\mathrm{JD, QD, KD, JH, QH, KH} \right \}, P(A \cap B) = \frac{6}{52} \approx 11.5 \%

A \cup B = \left \{ \mathrm{all \ 13 \ hearts, \ all \ 13 \ diamonds, \ JS, \ QS, \ KS, \ JC, \ QC, \ KC} \right \}. P(A \cup B) = \frac{32}{52} \approx 61.5 \%

2. A \cap B  = \left \{  \right \} = \mathrm{the \ empty \ set} = \varnothing, A \cup B = \left \{ \mathrm{M, T, W, H, F, S, U}\right \} = \mathrm{the \ sample \ space}

3. Answers will vary.

The Complement of an Event

Assignment: Just Two Possibilities

Being able to calculate a probability often depends on thinking of the situation in the right way, by correctly fitting it into a mold. It is frequently useful to take a sample space with many possible outcomes and simplify it into one event and its complement. This is a step toward applying a binomial distribution. Students will learn about distributions in the next chapter. Give them the opportunity to play with dividing the sample space into an event and its complement now.

Example:

1. Divide the sample space of the following experiment into one event and its complement.

Experiment: A shopper purchases a box of cereal.

Outcomes: S = \left \{ \mathrm{all \ existing \ types \ of \ cereal} \right \}

Answer: A = \left \{ \mathrm{shopper \ buy \ cereal \ with \ less \ than \ 10 \ grams \ of \ sugar \ per \ serving}\right \} and

A’ = \left \{\mathrm{shopper \ buys \ cereal \ with \ 10 \ or \ more \ grams \ of \ sugar \ per \ serving} \right \}

Exercises:

1. Find two more ways the sample space from the experiment described in the example can be divided into an event and its complement.

2. Find two ways in which to divide the sample space of the following experiment into an event and its complement.

Experiment: A card is drawn from a standard deck.

Outcomes: S = \left \{\mathrm{all \ 52 \ possible \ cards} \right \}

3. Find two ways in which to divide the sample space of the following experiment into an event and its complement.

Experiment: The weight category associated with the BMI (body mass index) of an adult participant.

Outcomes: S = \left \{\mathrm{Underweight, \ Normal, \ Overweight, \ Obese} \right \}

Conditional Probability

Extension: Using Conditional Probability to Calculate the Probability of Intersections

In this section students calculate conditional probabilities using the formula P(A|B) = \frac{P(A \cap B)}{P(B)}, P(B) \neq 0. Multiplying both sides of this formula by P(B) yields a way to find the probability of the intersection of two sets, P(A \cap B) = P (A|B) \cdot P(B), P(B) \neq 0. This is an extremely useful and widely used method for calculating probabilities.

Exercises:

1. If a certain study finds that the probability of having an accident each year given that the driver regularly speeds is 0.2, and that 60 \% of drivers regularly speed. What is the probability that a randomly selected driver regularly speeds and will be in an accident this year?

Answer: Let A be the event of having an accident sometime during the year, and S be the event of selected driver who regularly speeds, then P(A|S) = 0.2, and P(S) = 0.6. So the probability that a randomly selected driver speeds and is involved in an accident is

P(A \cap S) = P(A|S) * P(S) = 0.2 * 0.6 = 0.12

2. Teresa is having trouble deciding between two elective courses. She estimates that the probability of getting an A in Environmental Studies is \frac{3}{4} and the probability of getting an A in Psychology is \frac{2}{3}. If she decides which class to take by flipping a fair coin. What is the probability she finishes the year with an A in Psychology?

Answer:

\mathrm{P(Psychology \ and \ A)} & = \mathrm{P(A \ given \ Psychology)*P(Psychology)} \\& = \frac{2}{3} * \frac{1}{2} = \frac{1}{3}

3. Use conditional probability to calculate the probability of drawing two diamonds from a standard deck of cards?

Answer: Let D_1 be the event that the first card is a diamond and D_2 be the event that the second card is a diamond, then the probability that both cards are diamonds is

P(D_1 \cap D_2) = P(D_1) * P(D_1|D_2) = \frac{13}{52} * \frac{12}{51} = \frac{156}{2652} \approx .0588

Additive and Multiplicative Rules

Extension: Tests for Independence

Many of the theorems and rules of probability apply only to independent events, and it is not always a simple matter to determine if two events are independent. There are two widely used tests for independence.

Two events, A and B, are independent if P(A|B) = P(A) or P(B|A) = P(B).

Two events, A and B, are independent if P(A \cap B) = P(A)*P(B)

Exercises:

A company of 200 employees is considering a new health care plan. The following distribution shows the responses of all 200 employees based on the variables gender and opinion when they are ask their opinion on the new plan.

In Favor Against
Female 30 90 120
Male 8 72 80
38 162

1. Are the events, F = being female, and A = being against of the new health plan, independent? Justify your answer with both definitions of independence.

Answer: No, gender and opinion on the healthcare plan are dependent.

P(F|A) & = \frac{90}{160} = 0.5625 \ \ \text{and} \ \ P(F) = \frac{38}{200} = .19\\P(F \cap A) & = \frac{90}{200} = .45 \ \ \text{and} \ \ P(F)*P(A) = \frac{38}{200} * \frac{162}{200} = .1539

Topics for Discussion:

  1. What does this imply about the healthcare plan?
  2. If the probabilities in the respective definitions were approximately equal, would the events be almost independent? How close to equal do probabilities calculated form this type of data need to be for the events to be considered independent?

Basic Counting Rules

Discussion and Activity: Taking a Simple Random Sample

In theory, making random selections from a population to form a sample sounds quite simple, but in practice, designing a method where each member of a large group has an equally likely chance to be chosen is often quite difficult. The challenge is finding a good sampling frame. A sampling frame is the list of units from which the sample is drawn. It might be a telephone directory, but not every member of the population is listed. It is difficult to find a complete sampling frame.

Generating Random Numbers

Once the frame is formed, each unit on the list can be assigned a number. Units will then be selected with a random number generator. Most calculators and computers can select random numbers. On the TI-84 for example, this can be done by selecting MATH, then moving over to PRB, the fifth option is randInt(. Three numbers are required in the argument of this function. The first two are the range in which the user would like the random number to be, and the third indicates how many numbers are required.

Discussion Topics:

  1. How would you take a simple random sample that represents this class? The school?
  2. What sampling frames could be used to get s simple random sample of the residents of a city, college student, or mothers? What members of the population would not be included in the frame? How would the absence of these members affect the results of the survey?

Activities:

  1. Have the students take a simple random sample of some population in the school, perhaps the athletes, honor roll students, or drama participants. Each selected student should be asked a question or complete a short survey, just to make the process more interesting. The instructor may have to request information form the school office for the students. This can be assigned to small groups who report their method and findings to the class.
  2. The students can research sampling frames. What creative methods have been used? What are the strengths /drawbacks of these methods? What outstandingly bad methods have been used? What were the results? This can be assigned to small groups and the results presented in class.

One option is to assign the first activity to some groups and the second to others, or let the students choose which assignment they would like to complete.

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